Rasul Shafikov

The University of Western Ontario | UWO

It is shown that any smooth closed orientable manifold of dimension 2 ⁢ k + 1 {2k+1} , k ≥ 2 {k\geq 2} , admits a smooth polynomially convex embedding into ℂ 3 ⁢ k {\mathbb{C}^{3k}} . This improves by 1 the previously known lower bound of 3 ⁢ k + 1 {3k+1} on the possible ambient complex dimension for such embeddings (which is sharp when k = 1 {k=1}...

We discuss local polynomial convexity of real analytic Levi-flat hypersurfaces in [Formula: see text], [Formula: see text], near singular points.

It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible ambient complex dimension for such embeddings (which is sharp when $k=1$). It is further shown that the embeddi...

It is shown that a (singular) real analytic hypersurface $X\subset \mathbb C^n$ is locally polynomially convex at a point $p\in X$ if and only if $X$ is Levi-flat and $p$ is not a dicritical singularity of $X$.

Rationally convex topological embeddings of compact surfaces (closed or with boundary) into $\mathbb{C}^2$ are constructed.

It is proved that dicritical singularities of real analytic Levi-flat sets coincide with the set of Segre degenerate points.

We show that, for $k>1$, any $2k$-dimensional compact submanifold of $\mathbb{C}^{3k-1}$ can be perturbed to be polynomially convex and totally real except at a finite number of points. This lowers the known bound on the number of smooth functions required on every $2k$-manifold $M$ to generate a dense subalgebra of $\mathcal{C}(M)$. We also show t...

It is shown that the unit ball in ${\mathbb C}^n$ is the only complex manifold that can universally cover both Stein and non-Stein strictly pseudoconvex domains.

This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the Reflection Principle, the scaling method, and the Kobayashi-Royden metric. We sketch the proofs of certain principal results and discuss some recent...

We characterize dicritical singularities of real analytic Levi-flat sets in terms of Segre varieties. As an application we study the Segre envelope of such a set.

We establish an effective criterion for a dicritical singularity of a real analytic Levi-flat hypersurface. The criterion is stated in terms of the Segre varieties. As an application, we obtain a structure theorem for some class of currents in the nondicritical case.

We obtain results on the existence of complex discs in plurisubharmonically convex hulls of Lagrangian and totally real immersions to Stein manifolds.

We show that a Lagrangian inclusion in $\mathbb C^2$ with double transverse self-intersection points and standard open Whitney umbrellas is rationally convex. As an application we show that any compact surface $S$, except $S^2$ and $\mathbb RP_2$, admits a pair of smooth complex-valued functions $f_1$, $f_2$ with the property that any continuous co...

It is shown that a Lagrangian inclusion of a real surface in $\mathbb C^2$ with a standard open Whitney umbrella and double transverse self-intersections is rationally convex.

It is shown that the Levi foliation of a real analytic Levi-flat hypersurface extends to a $d$-web near a nondicritical singular point and admits a multiple-valued meromorphic first integral.

Using the analytic theory of differential equations, we construct examples of formally but not holomorphically equivalent real-analytic Levi nonflat hypersurfaces in $\CC{n}$ together with examples of such hypersurfaces with divergent formal CR-automorphisms.

We give a sufficient condition for a meromorphic correspondence to be a holomorphic correspondence in a neighbourhood of a smooth real hypersurface

We obtain local and global results on polynomially convex hulls of Lagrangian and totally real submanifolds of $C^n$ with self-intersections and open Whitney umbrella points.

We study the analytic continuation problem for a germ of a biholomorphic mapping from a non-minimal real hypersurface $M\subset\CC{n}$ into a real hyperquadric $\mathcal Q\subset\CP{n}$ and prove that under certain non-degeneracy conditions any such germ extends locally biholomorphically along any path lying in the complement $U\setminus X$ of the...

The paper considers a class of Lagrangian surfaces in $\mathbb C^2$ with isolated singularities of the unfolded Whitney umbrella type. We prove that generically such a surface is locally polynomially convex near a singular point of this kind.

An extension theorem for holomorphic mappings between two domains in $\mathbb C^2$ is proved under purely local hypotheses. Comment: 22 pages

Given a real analytic set X in a complex manifold and a positive integer d, denote by A(d) the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that A(d) is a closed semianalytic subset of X. Comment: 16 pages

A general class of singular real hypersurfaces, called subanalytic, is defined. For a subanalytic hypersurface M in C^n, Cauchy-Riemann (or simply CR) functions on M are defined, and certain properties of CR functions discussed. In particular, sufficient geometric conditions are given for a point p on a subanalytic hypersurface M to admit a germ at...

Given a real analytic (or, more generally, semianalytic) set R in the n-dimensional complex space, there is, for every point p in the closure of R, a unique smallest complex analytic germ X_p that contains the germ R_p. We call the complex dimension of X_p the holomorphic closure dimension of R at p. We show that the holomorphic closure dimension o...

Given a domain Y in a complex manifold X, it is a difficult problem with no general solution to determine whether Y has a schlicht envelope of holomorphy in X, and if it does, to describe the envelope. The purpose of this paper is to tackle the problem with the help of a smooth 1-dimensional foliation F of X with no compact leaves. We call a domain...

It is proved that a germ of a real analytic CR map from a smooth real-analytic minimal CR manifold M to an essentially finite real-algebraic generic submanifold M' of P^N of the same CR-dimension extends as a holomorphic correspondence along M. Applications are given for pseudoconcave submanifolds of P^N.

It is proved that CR functions on a quadratic cone M in C n, n> 1, admit one-sided holomorphic extension if and only if M does not have two-sided support, a geometric condition on M which generalizes minimality in the sense of Tumanov. A biholomorphic classification of quadratic cones in C 2 is also given. 1. Introduction. One of the central result...

If R is a real analytic set in \(\mathbb{C}^{n}\) (viewed as \(\mathbb{R}^{2n}\)), then for any point p∈R there is a uniquely defined germ X p of the smallest complex analytic variety which contains R p , the germ of R at p. It is shown that if R is irreducible of constant dimension, then the function p→ dim X p is constant on a dense open subset o...

In this paper we extend the results on analytic continuation of germs of holomorphic mappings from a real analytic hypersurface to a real algebraic hypersurface to the case when the target hypersurface is of higher dimension than the source. More precisely, we prove the following: Let M be a connected smooth real analytic minimal hypersurface in Cn...

It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two. In this brief note we use our uniformization result from [10, 11] to extend the work of Fu and Wong [7] on the relationship between two long-standing conjectures about the behaviour of the Bergman metric...

LetM, M′ be smooth, real analytic hypersurfaces of finite type in ℂn and [^(f)]\hat f a holomorphic correspondence (not necessarily proper) that is defined on one side ofM, extends continuously up toM and mapsM to M′. It is shown that [^(f)]\hat f must extend acrossM as a locally proper holomorphic correspondence. This is a version for corresponden...

It is shown that two strictly pseudoconvex Stein domains with real analytic boundaries have biholomorphic universal coverings provided that their boundaries are locally biholomorphically equivalent. This statement can be regarded as a higher dimensional analogue of the Riemann uniformization theorem.

In this paper we study the dynamics of regular polynomial automorphisms of C^n. These maps provide a natural generalization of complex Henon maps in C^2 to higher dimensions. For a given regular polynomial automorphism f we construct a filtration in C^n which has particular escape properties for the orbits of f. In the case when f is hyperbolic we...

In this note we derive an upper bound for the Hausdorff dimension of the stable set of a hyperbolic set $\Lambda$ of a $C^2$ diffeomorphisms on a $n$-dimensional manifold. As a consequence we obtain that $\dim_H W^s(\Lambda)=n$ is equivalent to the existence of a SRB-measure. We also discuss related results in the case of expanding maps.

Let f be a proper holomorphic mapping between bounded domains D and D' in C^2. Let M, M' be open pieces on the boundaries of D and D' respectively, that are smooth, real analytic and of finite type. Suppose that the cluster set of M under f is contained in M'. It is shown that f extends holomorphically across M. This can be viewed as a local versio...

The following result is proved: Let D and D′ be bounded domains in ℂn , ∂D is smooth, real-analytic, simply connected, and ∂D′ is connected, smooth, real-algebraic. Then there exists a proper holomorphic correspondence f:D→D′ if and only if there exist points p∈∂D and p′∈∂D′, such that ∂D and ∂D′ are locally CR-equivalent near p and p′. This leads...

We show that a proper holomorphic mapping from a domain with real-analytic boundary to a domain with real-algebraic boundary extends holomorphically to a neighborhood of .

Complex analysis and complex geometry can be viewed as two aspects of the same subject. The two are inseparable, as most work in the area involves interplay between analysis and geometry. The fundamental objects of the theory are complex manifolds and, more generally, complex spaces, holomorphic functions on them, and holomorphic maps between them....